Question

1. if the following graph is transformed so that it is 6 times as steep and is translated 4 units up, what is the new equation?

○ ( y = 6x + 4 )
○ ( y = \frac{1}{2}x + 6 )
○ ( y = 3x + 1 )
○ ( y = \frac{1}{2}x + 6 )

Explanation:

First, we assume the original graph is a linear equation in the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y - intercept.

Step 1: Analyze the steepness transformation

If the graph is 6 times as steep, we multiply the original slope \( m \) by 6. Let's assume the original slope was \( m = 1\) (for a simple case, since we are looking at the transformation). After making it 6 times as steep, the new slope \( m_{new}=6\times1 = 6\).

Step 2: Analyze the vertical translation

A translation of 4 units up means we add 4 to the original y - intercept. If we assume the original y - intercept \( b = 0\) (for a simple linear equation like \( y=x\)), after translating 4 units up, the new y - intercept \( b_{new}=0 + 4=4\).
So the new equation of the line (in slope - intercept form \( y=mx + b\)) is \( y = 6x+4\)

Answer:

\( y = 6x + 4\) (the first option among the given choices)